Davis' Convexity Theorem and Extremal Ellipsoids

TL;DR

This paper explores the uniqueness of minimal and maximal ellipsoids related to convex bodies, providing conditions for uniqueness and illustrating cases with non-unique solutions.

Contribution

It introduces new criteria based on convexity or concavity for the uniqueness of extremal ellipsoids in convex bodies.

Findings

Uniqueness results for circumscribing and inscribed ellipsoids

Examples demonstrating non-uniqueness cases

Criteria based on convexity/concavity for ellipsoid size functions

Abstract

We give a variety of uniqueness results for minimal ellipsoids circumscribing and maximal ellipsoids inscribed into a convex body. Uniqueness follows from a convexity or concavity criterion on the function used to measure the size of the ellipsoid. Simple examples with non-unique minimal or maximal ellipsoids conclude this article.